Farquhar, who pitched the sixth inning of the game, suffered a brain hemorrhage in the dugout afer the inning and was rushed to a local hospital.
Later testing revealed a ruptured aneurysm that caused his brain to bleed, the White Sox said in a statement.
He is currently in critical, but stable, condition at RUSH Hospital, the White Sox said. No further information on his condition was made available.
"The White Sox will provide additional updates on Farquhar's health over the coming days as appropriate, but the club also asks that you respect the privacy of the Farquhar family at this time," the team said.
The Astros, the team that is currently in Chicago playing the White Sox, sent a message of support to Farquhar on social media:
"Get well soon Danny," the tweet reads. "Our thoughts are with him and his family during this difficult time."
White Sox manager Rick Renteria offered his thoughts and prayers to his pitcher.
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"We all need to stay positive and be focused on thinking and praying for the best," he said before Saturday's game.
"It crushes us in this clubhouse," pitcher James Shields said. "Nothing really matters baseball-wise when something like that happens, you know?"
The Sox also asked for fans to offer their positive thoughts and prayers as Farquhar is treated.